Fluctuation Theory and Critical Phenomena

Abstract

Singularities in equilibrium properties at critical points are related to the asymptotic forms of averages of products of ν(r), the deviation from the mean density at position r. Thus the forms of the singularities of the thermodynamic properties at the critical point can be obtained in a simple fashion from any theory that describes the forms of correlation functions like 〈ν(r1)ν(r2) 〉 for large r12. As a particular example of such a theory we use the Landau–Lifshitz fluctuation theory to obtain the results Cυ∼κ−1, KT∼κ−2, and ∼(T − Tc)1/2κ for T near Tc. We demonstrate that the predictions of the fluctuation theory are consistent in the sense that the prediction of the form of Cυ from four-particle correlations is the same as that obtained from a temperature integration of ∂Cυ / ∂T in terms of six-particle corrections.